Decide whether the set S = {−1, 0 ,1} is closed under each of the following operations. Be sure to justify
each of your conclusions.

1. Multiplication
2. Addition
3. Subtraction
4. Division
5. Squaring
6. Square rooting
7. Averaging 2 numbers
8. Averaging 3 numbers

Answers

Answer 1

Answer:

Explained below

Step-by-step explanation:

Sol: 1. Multiplication

Multiply the set S; the answer should also belong to the set S. For example, -1 x 0 = 0 Є S and -1 x 1 = -1 Є S, Hence the set S is closed under multiplication.

2. Addition

When we add two sets S elements, the answer should also belong to the set S, for example, 1 + 0 = 1 Є S but 1 + 1 = 2, which does not belong to set S; hence S is not closed under addition.

3. Subtraction

The set S is not closed under subtraction because  -1 – 1  = -2, -2 is not the set S.

4. Division

The set S is not a closed division because when we divide one S element to another element, the answer does not belong to the set S, for example -1/0 = undefined.

5. Squaring

The set is closed under squaring if we square an element of the set S, and the result also belongs to the set S, for example, (0)2 = 0 and (-1)2 = 1, which belongs to the set S. Hence set S is closed under squaring.

6. Square rooting

The set S is not closed under square rooting because when we take the square root of an element of set S, the answer does not belong to the set S. √-1 = undefined.

7. Averaging two numbers

Averaging two numbers mean the sum of two numbers of set S is divided by two. For example, 1 + 0/2 = 1/2, which is not belongs to set S. hence set is not closed under averaging two numbers.

8. Averaging three numbers

The sum of three numbers of set S is divided by 3; the answer should belong to the set S.

-1 + 0 + 1 / 3= 0, 0 belongs to set. Hence the set S is closed under averaging three numbers.

Answer 2

Answer is explained below. We can say that Subtraction is not closed under the set.

Given set S = {−1, 0 ,1}  we have to check the given set is closed or not.

1. Multiplication

Multiply the set S; the answer should also belong to the set S. For example, [tex]-1\times 0=0[/tex], 0 Є S and [tex]-1\times 1 = -1[/tex] , -1 Є S, so the set S is closed under multiplication.

2. Addition

On adding the any two values of the set, the result belong to the set itself, for ex if we add [tex]-1+0=-1\\[/tex] Є S  but 1. so we can say that the given set is closed under addition.

3. Subtraction

The set S is not closed under subtraction because  -1 – 1  = -2, -2 is not the set element S.

4. Division

The set S is not belongs to closed division because when we divide one S element to another element, the answer does not belong to the set S, for example -1/0 = undefined.

5. Squaring

The set is closed under squaring,since we square an element of the set S, and the result also belongs to the set S. for example, [tex]0^{2} =0[/tex] and[tex]-1^{2} =1[/tex] which belongs to the set S. Hence set S is closed.

Hence We can say that Subtraction is not closed under the set.

For more details on Set follow the link below:

https://brainly.com/question/8053622


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Step-by-step explanation:

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Step-by-step explanation:

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Step-by-step explanation:

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Answers

Answer:

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Step-by-step explanation:

Solve the following system:

{3 x + 2 y - z = 8

2 x + 2 z = -4

x + 3 y = 4

Hint: | Choose an equation and a variable to solve for.

In the third equation, look to solve for x:

{3 x + 2 y - z = 8

2 x + 2 z = -4

x + 3 y = 4

Hint: | Solve for x.

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{3 x + 2 y - z = 8

2 x + 2 z = -4

x = 4 - 3 y

Hint: | Perform a substitution.

Substitute x = 4 - 3 y into the first and second equations:

{3 (4 - 3 y) + 2 y - z = 8

2 (4 - 3 y) + 2 z = -4

x = 4 - 3 y

Hint: | Expand the left hand side of the equation 3 (4 - 3 y) + 2 y - z = 8.

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{12 - 7 y - z = 8

2 (4 - 3 y) + 2 z = -4

x = 4 - 3 y

Hint: | Expand the left hand side of the equation 2 (4 - 3 y) + 2 z = -4.

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8 - 6 y + 2 z = -4

x = 4 - 3 y

Hint: | Choose an equation and a variable to solve for.

In the first equation, look to solve for y:

{12 - 7 y - z = 8

8 - 6 y + 2 z = -4

x = 4 - 3 y

Hint: | Isolate terms with y to the left hand side.

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8 - 6 y + 2 z = -4

x = 4 - 3 y

Hint: | Solve for y.

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x = 4 - 3 y

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{y = 4/7 - z/7

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{y = 4/7 - z/7

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x = 4 - 3 y

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{y = 4/7 - z/7

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Hint: | Isolate terms with z to the left hand side.

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{y = 4/7 - z/7

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x = 4 - 3 y

Hint: | Solve for z.

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{y = 4/7 - z/7

z = -3

x = 4 - 3 y

Hint: | Perform a back substitution.

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z = -3

x = 4 - 3 y

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x = 1

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Step-by-step explanation:

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Step-by-step explanation:

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Step-by-step explanation:

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Answers

Answer:

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Step-by-step explanation:

We are given;

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Formula for class width is;

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Answers

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Step-by-step explanation:

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Step-by-step explanation:

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Step-by-step explanation:

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